Vibration Control of Flexible Structures Using Semi-Active ...cdn.intechopen.com/pdfs/37568/InTech-Vibration_control_of_flexible... · Chapter 5 Vibration Control of Flexible Structures

Chapter 5

Vibration Control of Flexible Structures Using Semi-Active Mount : Experimental Investigation

Seung-Bok Choi, Sung Hoon Ha and Juncheol Jeon

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48823

1. Introduction

In many dynamic systems such as robot and aerospace areas, flexible structures have beenextremely employed to satisfy various requirements for large scale, light weight and highspeed in dynamic motion. However, these flexible structures are readily susceptible to theinternal/external disturbances (or excitations). Therefore, vibration control schemes shouldbe exerted to achieve high performance and stability of flexible structure systems. Recently,in order to successfully achieve vibration control for flexible structures smart materials suchas piezoelectric materials [1-2], shape memory alloys [3-4], electrorheological (ER) fluids[5-6] and magnetorheological (MR) fluids [7] are being widely utilized. Among these smartmaterials, ER or MR fluid exhibits reversible changes in material characteristics when sub‐jected to electric or magnetic field. The vibration control of flexible structures using thesmart ER or MR fluid can be achieved from two different methods. The first approach is toreplace conventional viscoelastic materials by the ER or MR fluid. This method is very effec‐tive for shape control of flexible structures such as plate [5]. The second approach is to de‐vise dampers or mounts and apply to vibration control of the flexible structures. Thismethod is very useful to isolate vibration of large structural systems subjected to externalexcitations [6-7]. In this work, a new type of MR mount is proposed and applied to vibrationcontrol of the flexible structures.

In order to reduce unwanted vibration of the flexible structure system, three different typesof mounts are normally employed: passive, semi-active and active. The passive rubber mount,which has low damping, shows efficient vibration performance at the non-resonant and highfrequency excitation. Thus, the rubber mount is the most popular method applied for vari‐ous vibrating systems. However, it cannot have a favorable performance due to small damp‐ing effect at the resonant frequency excitation. On the other hand, the passive hydraulic mount

© 2012 Choi et al.; licensee InTech. This is an open access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited.

has been developed to utilize dynamic absorber effect or meet large damping requirement inthe resonance of low frequency domain [8]. However, the high dynamic stiffness property ofthe hydraulic mount may deteriorate isolation performance in the non-resonant excitation do‐main. Thus, the damping and stiffness of the passive mounts are not simultaneously control‐lable to meet imposed performance criteria in a wide frequency range. The active mounts arenormally operated by using external energy supplied by actuators in order to generate con‐trol forces on the system subjected to excitations [9]. The control performance of the activemount is fairly good in a wide frequency range, but its cost is expensive. Moreover, its con‐figuration is complex and its stability may not be guaranteed in a certain operation condi‐tion. On the other hand, the semi-active mounts cannot inject mechanical energy into thestructural systems. But, it can adjust damping to reduce unwanted vibration of the flexiblestructure systems. It is known that using the controllable yield stress of ER or MR fluid, avery effective semi-active mount can be devised for vibration control of the flexible struc‐tures. The flow operation of the ER or MR mount can be classified into three different modes:shear mode [6], flow mode [10] and squeeze mode [11].

In this article, a new type of semi-active MR mount shown in the figure 1 is proposed and ap‐plied to vibration control of flexible structures. As a first step to achieve the research goal,the configuration of a mixed-mode MR mount is devised and the mathematical model is for‐mulated on the basis of non-dimensional Bingham number. After manufacturing an appro‐priate size of MR mount, the field-dependent damping force is experimentally evaluated withrespect to the field intensity. The MR mount is installed on the beam structure as a semi-ac‐tive actuator, while the beam structure is supported by two passive rubber mounts. The dy‐namic model of the structural system incorporated with the MR mount is then derived in themodal coordinate, and an optimal controller is designed in order to control unwanted vibra‐tion responses of the structural system subjected to external excitations. The controller is ex‐perimentally implemented and control performances such as acceleration of the structuralsystems are evaluated in frequency domain.

2. MR Mount

In this work, a new type of the mixed-mode MR mount which is operated under the flowand shear motion is proposed. The schematic configuration of the MR mount proposed inthis work is shown in Figure 1 (a). The MR mount consists of rubber element and MR dash-pot. The MR dash-pot is assembled by MR fluid, piston (or plunger), electromagnet coil, fluxguide, and housing. The MR fluid is filled in the gap between piston and outer cylindricalhousing. The electromagnetic coil is wired inside of the cylindrical housing. The housing canbe fixed to the supporting structure, and the plunger is attached to the top end of the rubberelement. The rubber element has a role to support the static load and isolate the vibrationtransmission at the non-resonant and high frequency regions. During the relative motion ofthe plunger and housing, MR fluid flows through annular gap. Thus, the pressure drop due

Advances in Vibration Engineering and Structural Dynamics88

to flow resistance of MR fluid in the annular gap can be obtainable. At the same time, theMR dash-pot has additional shear resistance due to relative motion of annular gap walls.Therefore, the proposed MR dash-pot operates under both the flow and shear modes. If nomagnetic field is applied, the MR dash-pot only produces a damping force caused by thefluid resistance associated with the viscosity of the MR fluid. However, if the magnetic fieldis applied through the annular gap, the MR mount produces a controllable damping forcedue to the yield stress of the MR fluid. As it can be seen from Figure 1 (a), the proposed MRmount has compact structure and operates without frictional components.

Figure 1. The proposed MR mount.

The transmitted force (Ft) through the proposed MR mount can be represented by

Ft =kr x + cr x + Fmr (1)

In the above, kr and cr are stiffness and damping of the rubber element, respectively. x is thedeflection of the rubber element. The damping force (Fmr) due to the flow resistance of theMR fluid in the gap can be represented by considering the plug flow of the MR fluid underthe mixed mode operation as follows [7].

Vibration Control of Flexible Structures Using Semi-Active Mount : Experimental Investigationhttp://dx.doi.org/10.5772/48823

89

φF =aφr3 + ( b

6 φc + c)φr2 (2)

where,

φF =Fmr

6πηvpL , φc =τyhηvp

, φr =rh (3)

In the above, φF is the dimensionless damping force, φc is the Bingham number, and φr isthe dimensionless geometric parameter. η is the post-yield plastic viscosity, vp is the relativevelocity between piston and housing, L is the annular gap length, τy is the field-dependentyield shear stress of the MR fluid, h is the annular gap size, and r is the piston radius. Thedimensionless parameter φc represents the ratio of the dynamic yield shear stress to the vis‐cous shear stress. Moreover, φc shows the influence of the magnetic field on the dampingforce of the MR mount. The dimensionless geometric parameter φr is characterized by thepiston radius r and gap size h . The dimensionless damping force φF increases as the Bing‐ham number φc and dimensionless geometric parameter φr increase. This implies that highyield stress and large piston area are required to generate high damping force of the MRmount. On the other hand, the gap length L depends on the dimensionless damping forceφF only. Therefore, the damping force Fmr can be directly scaled by the gap length L . It isnoted that the damping force Fmr can be expressed by the velocity of rubber element and thefield-dependent yield shear stress of the MR fluid. The parameters a and c of equation (2)are chosen to be 1 by considering Newtonian flow behavior of the MR fluid in the absence ofmagnetic field. And then, the parameter b , which reflects the effect of Bingham flow of theMR fluid, is chosen to be 2.47 using a least square error criterion.

The field-dependent yield stress τy of the MR fluid (MRF-132LD, Lord Corporation) em‐

ployed in this study has been experimentally obtained by 0.13H 1.3 kPa. Here the unit of mag‐netic field H is kA/m. The post-yield plastic viscosity η was also experimentally evaluatedby 0.59 Pasec. The rheometer (MCR300, Physica, Germany) was used for obtaining the val‐ue of the yield stress and the post-yield plastic viscosity. By considering both the nondimen‐sional damping force equations (2,3) and the size of the structural system, an appropriate sizeof MR mount was designed and manufactured as shown in Figure 1 (b). The piston radiusr , gap size h , and gap length L are designed to be 8.5 mm, 1.5 mm, and 10 mm, respective‐ly. The field-dependent damping force is predicted and experimentally measured by excit‐ing the MR mount with the sinusoidal signal which has frequency of 15 Hz and amplitudeof 0.07m/sec. Figure 2 presents controllable damping force characteristic of the proposed MRmount. In order to measure the field-dependent damping force of the MR mount, the MRmount is placed between the load cell and electromagnetic exciter. When the shaker tablemoves up and down by a command signal generated from the exciter controller, the MRmount produces the damping force and it is measured by the load cell. As expected, as the


input current increases, the damping force also increases. It is also observed that the predict‐ed field-dependent damping force shows small difference from the measured force in thepost-yield velocity region. On the other hand, the damping forces show much large differ‐ence in the pre-yield region. This is because the model (2,3) is discontinuous and cannot rep‐resent the hysteretic behavior of the MR mount in the transition from pre-yield to post-yield.

Figure 2. The field-dependent damping force of the MR mount.

The nondimensional form (2,3) of MR mount under mixed mode can be transformed to theBingham plastic model as follows [12]. This simple model is widely used for the controllerimplementations.

Fmr =cf (I )vp + Fy(I )sgn(vp) (4)

In the above, cf and Fy are the post-yield damping constant and the yield force, respectively.I is the current applied to the MR mount. In order to identify the parameters of the models,a constrained least-mean-squared (LMS) error minimization procedure is used. The costfunction J for Bingham plastic model is defined by

J (cf , Fy)=∑k=1

Nf (tk )− f (tk )

2(5)

where f (tk ) is the force calculated using the model given by equation (4), f (tk ) is the meas‐ured force shown in figure 2, and tk is the time at which the kth sample has been taken. Pa‐rameters cf and Fy of the Bingham plastic model are estimated so as to minimize the costfunction J . The parameter optimization is performed on a test case of the identification dataset. It is noted that the optimization procedure is applied to the data of the post-yield re‐gions for Bingham plastic model. Based on the measured response in Figure 2, the field-de‐pendent damping constant cf and yield force Fy are identified by (23 + 25I ) Nsec/m and


91

8.28I 1.85 N, respectively. Here, the unit of current I is A. These values will be used as systemparameters in the controller implementation for the structural system.

3. Structural System

In order to investigate the applicability of the proposed MR mount to vibration control, aflexible structure system is established as shown in Figure 3. The MR mount is placed be‐tween the exciting mass and steel beam structure. When the mass is excited by external dis‐turbance, the force transmitting through the MR mount excites the beam structure. Thus, thevibration of the beam structure can be controlled by activating the MR mount. y(pj) is thedisplacement at position pj of the structural system. The MR mount is placed at position p2

of the beam structure, while two passive rubber mounts are placed at p1 and p3 to supportthe beam structure. The governing equation of the motion of the proposed structural systemcan be obtained using the mode summation method as follows [13]:

q i(t) + 2ζiωiq i(t) + ωi2qi(t)=

Qi(t)I i

+Qexi(t)

I i, i =1, 2, ⋯ , ∞ (6)

where,

Qi(t)= (φi(p4)−φi(p2))(−Fmr(t)) (7)

Qe xi(t)=φi(p4)Fex(t) (8)

In the above, qi(t) is the generalized modal coordinate, φi(x) is the mode shape value at po‐sition x , ωi is the modal frequency, ζi is the damping ratio, and I i is the generalized mass ofthe i th mode. Qi(t) is the generalized force including the damping force of the MR mount,and Qexi(t) is the generalized force including the exciting force Fex(t) . It is noted that thespring and damping effects of the rubber elements are resolved in the modal parameters ofthe structural system.

In order to determine system parameters such as modal frequencies and mode shapes, mo‐dal analysis is undertaken by adopting a commercial software (MSC/NASTRAN for Win‐dows V4.0). The finite element model of the flexible structure consists of 30 beam elements,3 spring elements, and 2 mass elements. The nodes of the structural system are constrainedin the x, z directions, and the 2-node elastic beam element is used to model the beam. The ge‐ometry of the steel beam is 1500mm (length) 60mm (length) 15mm (thickness). The rubbermounts are placed on the 50mm ( p1 ) and 1450mm ( p3 ) from left end of the beam. The MRmount is placed on the 600mm ( p2 ) from the left end of the beam instead of the center ofthe beam. This is because that the MR mount located at center of the beam cannot control


the rotational modes. The 2-node spring elements are used to model the rubber mounts andrubber element of the MR mount. The material property for the spring elements is set by60kN/m which has been experimentally evaluated. To model the mass of lower part of theMR mount and vibrating mass, the 1-node lumped mass element is used. The material prop‐erties for the mass elements are 1.5kg and 3kg for the lower part of the MR mount and thevibrating mass, respectively.

Figure 3. Configuration of the structural system with the MR mount.

Figure 4 presents mode shapes of the first three modes. The first mode is bending mode, whilethe second and the third modes are combination of rotational and bending modes. In thisstudy, modal parameters of the structural system are identified by experimental modal test.It is evaluated by computer simulation that the effect of the residual modes is quite small com‐pared with dominant mode. Therefore, in this paper, only dominant modes are considered.

Figure 5 shows the schematic configuration of the experimental setup for the modal param‐eter identification. The mass on the MR mount is excited by the electromagnetic exciter. TheMR mount is removed for the modal parameter identification of structure system only. Thus,this test shows the modal test of the passive structural system in which the mass is support‐ed by the rubber mount. The excitation force and frequency are regulated by the exciter con‐troller. Accelerometers are attached to the mass and beam, and their positions are denotedby x = p1 , x = p2 , x = p3 and x = p4 . From this, the following system parameters are ob‐

tained : ω1 56.9 rad/s, ω2 144.8 rad/s, ω3 168.4 rad/s, ζ1 0.0260, ζ2 0.0497, and ζ3 0.0630.


93

Figure 4. Mode shapes of the structural system.

Figure 5. Experimental setup for the modal test of MR fluid mount system.

By considering first three modes as controllable modes, the dynamic model of the structural

system, given by equations (6-8), can be expressed in a state space control model as follows.

x(t)= Ax(t) + Bu(t) +w1 (9)

y(t)=Cx(t) +w2 (10)

where

x(t)= q1(t)q1(t)q2(t)q2(t)q3(t)q3(t) T (11)


y(t)= y(p4, t) y(p2, t) T (12)

u(t)= Fmr(t) (13)

A=

0 1 0 0 0 0−ω1

2 −2ζ1ω1 0 0 0 00 0 0 1 0 00 0 −ω2

2 −2ζ2ω2 0 00 0 0 0 0 10 0 0 0 −ω3

2 −2ζ3ω3

(14)

B =

0(φ1(p2)−φ1(p4)) / I1

0(φ2(p2)−φ2(p4)) / I2

0(φ3(p2)−φ3(p4)) / I3

(15)

C =0 φ1(p4) 0 φ2(p4) 0 φ3(p4)

0 φ1(p2) 0 φ2(p2) 0 φ3(p2)(16)

In the above, x(t) is the state vector, y(t) is the measured output vector, A is the systemmatrix, B is the control input matrix, C is the output matrix, and u(t) is the control input.w1 and w2 are uncorrelated white noise characterized by covariance matrices V1 and V2 asfollows [14].

Cov(w1, w1T )=V1

Cov(w2, w2T )=V2

Cov(w1, w2T )=0

(17)

4. Vibration Control

In order to attenuate unwanted vibration of the flexible structural system, the linear quad‐ratic Gaussian (LQG) controller, consisting of linear quadratic regulator (LQR) and Kalman-Bucy filter (KBF), is adopted. As a first step to formulate the LQG controller the performanceindex (J ) to be minimized is given by


95

J = ∫0∞{x T (t)Qx(t) + u T (t)Ru(t)}dt (18)

In the above, Q is the state weighting semi-positive matirx and R is the input weighting pos‐itive constant. Since the system (A, B) in equation (9) is controllable, the LQR controller forthe MR mount can be obtained as follows.

u(t)= −P −1B T x(t)= −Kx(t) (19)

In the above, K is the state feedback gain matrix, and P is the solution of the following alge‐braic Riccati equation :

A T P + PA−PBR −1B T P + Q =0 (20)

Thus, the control force of the MR mount can be represented as

u(t)= −Kx(t)= − (k1q1(t) + k2q1(t) + k3q2(t) + k4q2(t) + k5q3(t) + k6q3(t)) (21)

In this work, by the tuning method the control gains of the k1 , k2 , k3 , k4 , k5 and k6 are select‐ed by -536068, 8315.9, -327595, 3706.3, -15.99 and 2359.4, respectively. On the other hand, theproposed MR mount is semi-active. Therefore, the control signal needs to be applied accord‐ing to the following actuating condition [15]:

u(t)=u(t)

0, foru(t)( y(p4, t)− y(p2, t))0, foru(t)( y(p4, t)− y(p2, t))≤0

(22)

This condition physically indicates that the actuating of the controller u(t) assures the incre‐ment of energy dissipation of the stable system.

Since the states of qi(t) and q i(t) of the LQR controller (21) are not available from directmeasurement, the Kalman-Bucy filter (KBF) is formulated. The KBF is a state estimatorwhich is optimally considered in the statistical sense. The estimated state, x(t) , can be ob‐tained from the following state observer [14].

x(t)= Ax(t) + Bu(t) + L (y(t)−C x(t)) (23)

where,

L =OC TV2−1 =

l11

l21

l12

l22

⋯⋯

l16

l26

T(24)


In the above, L is the observer gain matrix, and O is the solution of the following observer

Riccati equation :

AO +OA T −OC TV2−1C P + V1 =0 (25)

Using the estimated states, the control force of the MR mount is obtained as follows.

u(t)= − (k1q1(t) + k2q1(t) + k3q2(t) + k4q2(t) + k5q3(t) + k6q3(t)) (26)

Figure 6. Experimental setup for vibration control.


97

Figure 7. Displacement response of the structural system with the MR mount.

Figure 8. Acceleration response of the structural system with the MR mount.

In order to implement the LQG controller, an experimental setup is established as shown inFigure 6. The mass supported by the MR mount is excited by the electromagnetic exciter, andthe excitation force and frequency are regulated by the exciter controller. Accelerometers are


attached to the beam and mass, and their positions are denoted by (x = p1) , (x = p2) ,

(x = p3) , and (x = p4) . Two accelerometers, attached on the positions and , are used forthe feedback signals. Velocity signals at these positions are obtained by using integrator cir‐cuit of charge amplifier. And the other accelerometers are used to measure vibration of thestructural system. The velocity signal which denoted by dashed line in Figure 6 is fed back tothe DSP (digital signal processor) via an A/D(analog to digital) converter. The state variablesrequired for the LQR (26) are then estimated by the KBF (23), and control voltage is deter‐mined by means of the LQR controller in the DSP. The control current is applied to the MRmount via a D/A (digital to analog) converter and a current supplier. The sampling rate in thecontroller implementation is set by 2kHz.

Figures 7 and 8 present the measured displacement and acceleration of the structural sys‐tem. The amplitude of excitations force is set by 1N. The excitation frequency range for thestructural system is chosen from 5 to 80Hz. The uncontrolled and controlled responses aremeasured at the positions of the mass () and beam structure (, ). It is observed from Fig‐ures 7 and 8 that the resonant modes of the structural system have been reduced by control‐ling the damping force of the MR mount. Especially, the 2nd and 3rd modes vibration at theposition has been substantially reduced by activating the MR mount. It is noted that theuncontrolled response is obtained without input current.

5. Conclusion

A new type of the mixed-mode MR mount was proposed and applied to vibration control of aflexible beam structure system. On the basis of non-dimensional Bingham number, an appro‐priate size of the MR mount was designed and manufactured. After experimentally evaluat‐ing the field-dependent damping force of the MR mount, a structural system consisting of aflexible beam and vibrating rigid mass was established. The governing equation of motion ofthe system was formulated and a linear quadratic Gaussian (LQG) controller was designed toattenuate the vibration of the structural system. It has been demonstrated through experimen‐tal realization that the imposed vibrations of the structural system such as acceleration anddisplacement are favorably reduced by activating the proposed MR mount associated with theoptimal controller. The control results presented in this study are quite self-explanatory justi‐fying that the proposed semi-active MR mount can be effectively utilized to the vibration con‐trol of various structural systems such as flexible robot arm and satellite appendages.

Acknowledgements

This work was partially supported by the National Research Foundation of Korea (NRF)grant funded by the Korea government (MEST)(No. 2010-0015090).


99

Author details

Seung-Bok Choi*, Sung Hoon Ha and Juncheol Jeon

*Address all correspondence to: [email protected]

Smart Structures and Systems Laboratory, Department of Mechanical Engineering, InhaUniversity, Incheon 402-751,, Korea

References

[1] Crawley, E. F., & de Luis, J. (1987). Use of Piezoelectric Actuators as Elements of In‐telligent Structures. AIAA Journal, 25(10), 1373-1385.

[2] Choi, S. B., & Kim, M. S. (1997). New Discrete-Time Fuzzy-Sliding Mode Controlwith Application to Smart Structures. AIAA Journal of Guidance, Control and Dynamics,20(5), 857-864.

[3] Roger, C. A. (1990). Active Vibration and Structural Acoustic Control of Shape Mem‐ory Alloy Hybrid Composites: Experimental Results. Journal of Acoustical Society ofAmerica, 88(6), 2803-2811.

[4] Choi, S. B., & Cheong, C. C. (1996). Vibration Control of a Flexible Beam Using SMAActuators. AIAA Journal of Guidance, Control, and Dynamics, 19(5), 1178-1180.

[5] Choi, S. B., Park, Y. K., & Jung, S. B. (1999). Modal Characteristics of a Flexible SmartPlate Filled with Electrorheological Fluids. AIAA Journal of Aircraft, 36(2), 458-464.

[6] Choi, S. B. (1999). Vibration Control of a Flexible Structure Using ER Dampers.ASME Journal of Dynamic Systems, Measurement and Control, 121(1), 134-138.

[7] Hong, S. R., & Choi, S. B. (2003). Vibration Control of a Structural System using Mag‐neto-rheological Fluid Mount. Proceedings of the 14th International Conference on Adap‐tive Structures and Technologies, 182-194, (Seoul, Korea).

[8] Ushijima, T. K., Takano, K., & Kojima, H. (1988). High Performance Hydraulic Mountfor Improving Vehicle Noise and Vibration. SAE Technical Paper Series. 880073.

[9] Tanaka, N., & Kikushima, Y. (1998). Optimal Design of Active Vibration IsolationSystems. Transactions of the ASME, Journal of Vibration, Acoustics, Stress and Reliabilityin Design, 110(1), 42-48.

[10] Hong, S. R., Choi, S. B., & Han, M. S. (2002). Vibration Control of a Frame Structureusing Electrorheological Fluid Mounts. International Journal of Mechanical Sciences,44(10), 2027-2045.


[11] Williams, E. W., Rigby, S. G., Sproston, J. L., & Stanway, R. (1993). ElectrorheologicalFluids Applied to an Automotive Engine Mount. Journal of Non-Newtonian Fluid Me‐chanics, 47, 221-238.

[12] Hong, S. R., Choi, S. B., Choi, Y. T., & Wereley, N. M. (2003). Comparison of Damp‐ing Force Models for an Electrorheological Fluid Damper. International Journal of Vehi‐cle Design, 33(1-3), 17-35.

[13] Meirovitch, L. (1990). Dynamics and Control of Structures. John Wiley & Sons Inc.

[14] Machejowski, J. M. (1989). Multivariable feedback design. Addison-Wesley Publish‐ing Company.

[15] Karnopp, D. (1990). Design Principles for Vibration Control Systems using Semi-ac‐tive Dampers. Journal of Dynamic Systems, Measurement and Control, 112(3), 448-455.


101

Documents

Vibration Control of Flexible Structures Using Semi-Active ...cdn.intechopen.com/pdfs/37568/InTech-Vibration_control_of_flexible... · Chapter 5 Vibration Control of Flexible Structures